In a gauge theory physical quantities are
gauge invariant: they do not depend on the gauge potential but on
the gauge equivalence class of the potential given.
The Seiberg-–Witten map relates the noncommutative gauge fields to the
commutative ones by requiring the noncommutative fields to have the
same gauge equivalence classes as the commutative ones
[1]}.
Explicitly, the noncommutative gauge
potential \(\hat{A}=A_\mu {\rm d}x^\mu \)  and the noncommutative gauge
parameters \({\hat{\epsilon }}\)  depend on the ordinary \(A\)  and \(\epsilon \)  so to satisfy:
\({\hat{A}}(A + \delta _\epsilon A) = {\hat{A}}(A) + {\hat{\delta }}_{\hat{\epsilon }}{\hat{A}}(A) \) 
